Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C82 · Schur multiplier (H2) = C25×C8 |
| direct product | C2×D3×C82 |
| Fitting | C2×C3×C82 ⋊ C2 |
| derived, Sylow | C3 ⋊ C22×C82 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C82 | D3 |
| commutator, hypocenter | C3 | C22×C82 |
| Frattini | C42 | C23×D3 |
| Fitting | C2×C3×C82 | C2 |
| socle | C23×C3 | C2×C42 |
| prime radical | C23×D3 | C42 |
| cosocle | C3×C42 | C24 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C82 | |
1342 conjugacy classes of subgroups; only the 811 normal subgroups are shown.
This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 811 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
384 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D3×C8×M32, D3×#7846, D3×#11402, … |
| maximal normal subgroup of | D3×C8×M32, D3×#7846, D3×#11402, … |
| maximal quotient of | D3×C8×M32 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 135133);